Q64E Consider the linear system.聽x1+... [FREE SOLUTION]

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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 2: Q64E (page 58)

Consider the linear system.
$\begin{matrix} x_{1} + 2 x_{2} + x_{3} + 7 x_{4} = 0 \\ x_{1} + 2 x_{2} + 2 x_{3} + 11 x_{4} = 0 \\ x_{1} + 2 x_{2} + 3 x_{3} + 15 x_{4} = 0 \\ x_{1} + 2 x_{2} + 4 x_{3} + 19 x_{4} = 0 \end{matrix}$
Find the matrix Bsuch that role="math" localid="1664193807311" $[\begin{array}{l} x_{1} \\ x_{3} \end{array}] = B [\begin{array}{l} x_{2} \\ x_{4} \end{array}]$ . See Exercise 63.

Short Answer

Expert verified

The matrix, $B = [\begin{matrix} 鈭� 2 & 鈭� 3 \\ 0 & 鈭� 4 \end{matrix}]$ .

Step by step solution

Consider the linear system

The system of equations is,

$\begin{matrix} x_{1} + 2 x_{2} + x_{3} + 7 x_{4} = 0 \\ x_{1} + 2 x_{2} + 2 x_{3} + 11 x_{4} = 0 \\ x_{1} + 2 x_{2} + 3 x_{3} + 15 x_{4} = 0 \\ x_{1} + 2 x_{2} + 4 x_{3} + 19 x_{4} = 0 \end{matrix}$

The matrix form of the equations is,

$[\begin{matrix} 1 & 2 & 1 & 7 & 0 \\ 1 & 2 & 2 & 11 & 0 \\ 1 & 2 & 3 & 15 & 0 \\ 1 & 2 & 4 & 19 & 0 \end{matrix}]$

Compute the matrix

The row-echelon form of the matrix is,

$\begin{matrix} [\begin{matrix} 1 & 2 & 1 & 7 & 0 \\ 1 & 2 & 2 & 11 & 0 \\ 1 & 2 & 3 & 15 & 0 \\ 1 & 2 & 4 & 19 & 0 \end{matrix}] ~ [\begin{matrix} 1 & 2 & 1 & 7 & 0 \\ 0 & 0 & 1 & 4 & 0 \\ 0 & 0 & 3 & 12 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{matrix}] \\ ~ [\begin{matrix} 1 & 2 & 0 & 3 & 0 \\ 0 & 0 & 1 & 4 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{matrix}] \end{matrix}$

Find the matrix

Represent the equation in terms of a matrix.

$\begin{matrix} x_{1} + 2 x_{2} + 3 x_{4} = 0 \\ 鈬� x_{1} = 鈭� 2 x_{2} 鈭� 3 x_{4} \\ x_{3} + 4 x_{4} = 0 \\ 鈬� x_{3} = 鈭� 4 x_{4} \end{matrix}$

The matrix is,

$\begin{matrix} [\begin{array}{l} x_{1} \\ x_{3} \end{array}] = B [\begin{array}{l} x_{2} \\ x_{4} \end{array}] \\ 鈬� [\begin{array}{l} x_{1} \\ x_{3} \end{array}] = [\begin{matrix} 鈭� 2 & 鈭� 3 \\ 0 & 鈭� 4 \end{matrix}] [\begin{array}{l} x_{2} \\ x_{4} \end{array}] \\ 鈭� B = [\begin{matrix} 鈭� 2 & 鈭� 3 \\ 0 & 鈭� 4 \end{matrix}] \end{matrix}$

Hence, the matrix, $B = [\begin{matrix} 鈭� 2 & 鈭� 3 \\ 0 & 鈭� 4 \end{matrix}]$ such that $[\begin{array}{l} x_{1} \\ x_{3} \end{array}] = B [\begin{array}{l} x_{2} \\ x_{4} \end{array}]$ .

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Short Answer

Step by step solution

Consider the linear system

Compute the matrix

Find the matrix

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91影视

Consider the linear system.x1+2x2+x3+7x4=0x1+2x2+2x3+11x4=0x1+2x2+3x3+15x4=0x1+2x2+4x3+19x4=0Find the matrix Bsuch that role="math" localid="1664193807311" [x1x3]=B[x2x4]. See Exercise 63.

Short Answer

Step by step solution

Consider the linear system

Compute the matrix

Find the matrix

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Pure Maths

Probability and Statistics

Geometry

Statistics

Decision Maths

Study anywhere. Anytime. Across all devices.